What is the condition for the lines represented by the equation 2x^2 + 3xy + y^2 = 0 to be coincident?

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Q: What is the condition for the lines represented by the equation 2x^2 + 3xy + y^2 = 0 to be coincident?

Solution: The lines are coincident if the discriminant D of the quadratic equation is zero.

Steps: 7

Step 1: Understand that the equation 2x^2 + 3xy + y^2 = 0 represents a pair of lines.
Step 2: Identify that for two lines to be coincident, they must be exactly on top of each other.
Step 3: Recall that the condition for two lines represented by a quadratic equation to be coincident is that the discriminant (D) must be zero.
Step 4: Calculate the discriminant D using the formula D = B^2 - 4AC, where A, B, and C are the coefficients from the quadratic equation.
Step 5: In our equation, A = 2, B = 3, and C = 1. Substitute these values into the discriminant formula: D = (3)^2 - 4(2)(1).
Step 6: Simplify the expression: D = 9 - 8 = 1.
Step 7: Since D is not zero (D = 1), the lines are not coincident.